1 Residual error energy
CLS LS
Fig. 3.44 Energy of the residual error ϵ. Results for both LS and CLS method.
Spacing of the DLLds= 0.2 with an output every Ta= 200 ms.
3.5.1 LS and CLS comparison
To compare the performance in terms of Pd and Pf a of the detectors, we used the same dataset used in ??. The results are illustrated in Fig. 3.45 and Fig.
3.46 for different value of α. In Fig. 3.45 the results are related to LAF-LS, while Fig. 3.46 to LAF-CLS solution. It is possible to observe that the two detectors, in terms of Pd, have similar performance, instead, the situation is different in terms of Pf a. As we already mentioned, the reason why we choose a constraint is because the noise at the input of theLAF-LS generates possible false alarm, especially in case of low C/N0.
By using these plots, it is even possible to select values for α to tune the entries of the dictionary for the detector.
3.6 Statistical plots for the coefficients: Q-Q plot and mountain plot
Here, we present two graphical tools used in the work. The first, the mountain plot, also called folded empirical Cumulative Density Function (CDF) plot, presented in [55], is a graphical tool alternative to the CDF. The CDF is a strictly increasing function, the mountain plot, instead, folds the second half
Fig. 3.45 Pd(upper graph) and Pf a (bottom graph) for different value of α in case ofLS decomposition.
Fig. 3.46 Pd(upper graph) and Pf a (bottom graph) for different value of α in case ofCLS decomposition.
3.6 Statistical plots for the coefficients: Q-Q plot and mountain plot 93
of CDF. Therefore the resulting plot is identical to the CDFfor the elements X with probability P (X < x) ≤ 0.5 while, it is plotted 1 − P (X ≤ x) for the elements X with probability P (X > x) > 0.5. This instrument is quite useful to highlight some properties, such as symmetry, and to emphasize the median of the distribution [56]. It is also useful and easier to find the central amount of percentage (i.e 95%) of the data. Finally, different distributions can be compared more easily.
Another graphical tool used is the Q-Q plot (stands for Quantile-Quantile plot) to compare two probability distributions by using quantiles against each other. If the two distributions are similar, the Q-Q plot will be a line y = x.
In our case Q-Q plot is used to compare an empirical distribution against a Normal distribution.
To populate the dictionary used for the detection, we can start by evaluating the distribution of the coefficients in case of LAFwith LS andCLS solution.
First of all, we started with a simulation scenario with only noise with a duration of 30 minutes and a moving average window of 500 ms. The case of LAF-LS is in Fig. 3.47, where it is possible to see a good symmetry for all the M = 9 distributions. It is plotted also a Gaussian distribution (the circle marker) that overbound the 5-th coefficient, the center one. As already explained in the chapter, it is the coefficient representing the LOS in the ideal case. Its distribution is centered around a value different from zero. The other distributions, in case of only noise, are centered around zero. Instead, others distributions of the coefficients are centered around zero and have different variances. It is still possible to overbound with a zero mean Gaussian distribution and a standard deviation larger (σ = 0.5) than the center coefficient distribution (σ = 0.2). Another observation regards the distribution of the 4-th and 6-th coefficient. As in the case of the 5-th coefficient, they have a bias, since they are not centered around zero. This depends on the decomposition mechanism of filter together with the position of the peak leading to a not perfect symmetry of the measured correlation. This is more visible in case of LAF-CLS solution as in Fig. 3.48. The Q-Q plot in Fig. 3.49 is an example, for the distributions centered around zero, of the similarity with a Normal distribution in a limited range.
-1 -0.5 0 0.5 1 1.5 2 2.5
Amplitude
10-3 10-2 10-1
Probability
Mountain plot of the coeffs.
1 2 3 4 5 6 7 8 9 Gaussian Dist.
Fig. 3.47 Distribution of all M coefficients of the filter in case ofLAF-LS solution with C/N0= 45 dBHz.
In Fig. 3.48 a mountain plots in the case ofLAF-CLS is shown. Here it is possible to observe the huge separation between the zero-centered distributions and the LOS distribution. This depends on the fact that in the same noise condition, the variances of the non-LOS components is smaller than in the LAF-LS case.
3.6.1 Analogy with Transmission Theory
As already said, the analysis over the incoming signals were performed at the tracking stage of the GNSS receiver. The correlation signal obtained was decomposed by solving aLSor CLSminimization problem. The decomposition works not directly on the signal, but on the shape of the correlation. All the considerations about the detector are based on the vector of the filter coefficients in a vector space domain. The decision metric chosen is the minimum euclidean distance between vectors in a geometrical space. By using a multicorrelator and
3.6 Statistical plots for the coefficients: Q-Q plot and mountain plot 95
0 0.5 1 1.5 2 2.5
Amplitude
10-4 10-3 10-2 10-1 100
Probability
Mountain plot of the coeffs.
1 2 3 4 5 6 7 8 9
Fig. 3.48 Distribution of all M coefficients of the filter in case ofLAF-CLS solution with C/N0= 45 dBHz.
-4 -3 -2 -1 0 1 2 3 4 Standard Normal Quantiles
-1.5 -1 -0.5 0 0.5 1 1.5
Quantiles of Input Sample
QQ Plot of 9-th coeff. vs Standard Normal
Fig. 3.49 Q-Q plot of the 9-th coefficient in case ofLAF-LS solution with C/N0= 45 dBHz.
the decomposition, we could have more information about theMP, so the reason why we chose to use a dictionary of vectors for comparisons. This approach leads to an analogy with another communications topic: the transmission theory [57]. By transforming the problem from signal to geometrical domain, we have similar conditions as in case of reception of a symbol, on an AWGN channel, and try to guess which symbol of the constellation we received.
In transmission theory, in case of the classical modulation scheme like BPSK, Pulse Amplitude Modulation (PAM), Quadrature Amplitude Modu-lation (QAM) etc., we have a set (s1, s2, . . . , si, . . . , sn), called constellation, of the possible symbol that can be received. The constellation can be represented, for example, in a plane and the symbols are points distributed in that space.
The received symbol sr, due to the noise or other impairments, is a point in the space that probably has coordinates not coincident with the coordinates of the constellation symbols. To decide which symbol was transmitted, the space is divided in regions, called Voronoi regions, around the symbols of the constellation. The decision taken is that the incoming symbol sr is a symbol of the constellation (s1, s2, . . . , si, . . . , sn) if it is within one of these regions.